We have analysed the effects of the dissipation terms on numerical stability when approximated on the explicit side or on the implicit side using a model equation. Now let us do the same analysis for the convection terms.

Statement 2: In terms of numerical stability, it is better to approximate the convection terms on the implicit side (LHS) rather than to approximate them on the explicit side (RHS).

In order to explain this statement let us use another simple equation as the following

(48)

If we prefer to approximate the convection term on the explicit side we will get

(49)

When we apply a numerical stability analysis to this equation, we find that this scheme is numerical unstable

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Von Neuman Stability Analysis of Explicit 1-D Convection Equation:

We have the following equation

(50)

The finite difference equation will be as the following

(51)

or

(52)

where

(53)

Let us define

(54)

Then

(55)

Substituting these into (52) we get

(56)

cancelling out some terms we get

(57)

Let

(58)                   and

Since

(59)

Hence

(60)

Note that is imaginary and the amplitude ( ) is equal to

(61)

The amplification factor for any value of . Therefore the numerical scheme is unconditionally unstable.

Reference:

Tannehill, Anderson and Pletcher, Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, page 112 (last paragraph)

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On the other hand if go back to equation (48) and approximate the convection term on the implicit side we will get

(62)

In operator notation the above equation looks like the following

(63)

If we apply a numerical stability analysis we see that this equation is unconditionally stable.

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Von Neuman Stability Analysis of Implicit 1-D Convection Equation:

We have the following equation

(64)

With operator notation this equation looks like the following

(65)

The finite difference equation will look like

(66)

or

(67)

where

(68)

Let

(69)

Then

(70)

Substituting these into (67) we get

(71)

When we cancel out some terms

(72)

and define

(73)                   and

and also use the following trigonometric definition

(74)

we get the following

(75)

Hence

(76)

or

(77)

Note that is complex and the amplitude ( ) is equal to

(78)

Therefore the amplitude of the amplification factor is for any value of . There is one thing I would like to mention here, note that can be negative or positive depending on the value of . However since there is in equation (78), the numerical scheme is unconditionally stable regardless of having a negative or positive value.

Reference:

Tannehill, Anderson and Pletcher, Computational Fluid Mechanics and Heat Transfer, Taylor&Francis, page 113-115

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Let us rewrite equation (71)

(79)

Rearranging this equation we get

(80)

Since and , substituting we get

(81)

Now let us rewrite equation (63)

(82)

From equations (81) and (82), we can consider the LHS operator as a constant obtained after a Fourier expansion. In this case the operator is equivalent to

(83)

We see that if the convection term is approximated on the explicit side the numerical scheme is unstable, however when we approximate the convection term on the implicit side we see that the scheme is unconditionally stable regardless of u being negative or positive.

Note that, the convection terms we talk about have as a coefficient. Therefore the convection terms, we examined on either RHS or LHS, are first order () terms. This information will be usefull in the next pages.